Relay should work , but the LED, not so much… The LED in that cct is the wrong way around.
Transistor is wired OK – it is an NPN, so diagram is correct.
Also, the resistor from the I/O pin ( 13) to the base is very low in ohm – The poor arduino has to provide around 10mA to drive the base( if the arduino is a 3.3v job, else 20mA if its 5v) – 2k2 ohm is quite adequate.
So, I am dismayed that the led flashes in this circuit – it cannot when reverse biased…
Likewise dismayed that the relay does not activate, it should..unless the relay needs a higher voltage. What voltage does the relay coil require? Does it activate when you apply your 9v across the coil? If so..
Correct the LED connection direction
Remove the resistor from the arduino pin 13 and momentarily connect it to the arduino 3.3v or 5v supply.
The relay should activate. If it does not, did the LED come on?
If not, the voltage on the collector should be 9v with the base resistor unconnected. When connected to 3.3v/5v the collector voltage should be close to 0v. if not, transistor is dud..